That is a fallacy of affirming the antecedent (I think). It’s not enough to say "if X is true then we predict Y; we see Y therefore X is true." This approach fails to acknowledge any other possible explanations for Y.

All you can say logically is that if Y is true then X MAY be true.

Another very obvious explanation for the scenario you offer is that DNA sequences are similar because the animals have the same designer.

With the logic you offer, one can prove anything: flat Earth, life is a simulation, or red is blue.